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1. A single die is rolled. Find the probabilities of the following results, and give your answers as reduced fractions. Show your work and justify your answers. Use the formula sheet for reference of various formulas.  a. Are the events "<5" and "even" independent?

1. A single die is rolled. Find the probabilities of the following results, and give your answers as reduced fractions. Show your work and justify your answers. Use the formula sheet for reference of various formulas.  a. Are the events "<5" and "even" independent?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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1. A single die is rolled. Find the probabilities of the following results, and give your answers as reduced fractions. Show your work and justify your answers. Use the formula sheet for reference of various formulas. 

a. Are the events "<5" and "even" independent?

### Probability and Statistics Reference Sheet #### Basic Probability Principle \[ P(E) = \frac{n(E)}{n(S)} \] #### Union Rule - **For sets:** \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] - **For probability:** \[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \] #### Product Rule \[ P(E \cap F) = P(E) \cdot P(F|E) = P(F) \cdot P(E|F) \] #### Complement Rule (Probability) - \[ P(E') = 1 - P(E) \] - \[ P(E) = 1 - P(E') \] #### Complement Rule (Number of Outcomes) - \[ n(E) = n(S) - n(E') \] - \[ n(E') = n(S) - n(E) \] #### Bayes' Theorem \[ P(F|E) = \frac{P(E|F) \cdot P(F)}{P(E)} \] \[ P(E|F) = \frac{P(F|E) \cdot P(E)}{P(F|E) \cdot P(E) + P(F'|E) \cdot P(E') + \ldots + P(F) \cdot P(E|F)} \] #### Special Case for Bayes' Theorem \[ P(F|E) = \frac{P(F|E) \cdot P(E)}{P(F|E) \cdot P(E) + P(F'|E) \cdot P(E')}\] #### Permutations \[ P(n, k) = \frac{n!}{(n-k)!} \] #### Distinguishable Permutations \[ \frac{n!}{n_1! n_2! \ldots n_r!} \] #### Combinations \[ C(n, k) = \frac{n!}{k!(n-k)!} \] #### Conditional Probability \[ P(E|F) = \frac{n(E \cap F)}{n(F)} = \frac{P(E \cap F)}{P(F)} \] #### Number of Subsets - If a set has \( n \) elements, it
Transcribed Image Text:### Probability and Statistics Reference Sheet #### Basic Probability Principle \[ P(E) = \frac{n(E)}{n(S)} \] #### Union Rule - **For sets:** \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] - **For probability:** \[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \] #### Product Rule \[ P(E \cap F) = P(E) \cdot P(F|E) = P(F) \cdot P(E|F) \] #### Complement Rule (Probability) - \[ P(E') = 1 - P(E) \] - \[ P(E) = 1 - P(E') \] #### Complement Rule (Number of Outcomes) - \[ n(E) = n(S) - n(E') \] - \[ n(E') = n(S) - n(E) \] #### Bayes' Theorem \[ P(F|E) = \frac{P(E|F) \cdot P(F)}{P(E)} \] \[ P(E|F) = \frac{P(F|E) \cdot P(E)}{P(F|E) \cdot P(E) + P(F'|E) \cdot P(E') + \ldots + P(F) \cdot P(E|F)} \] #### Special Case for Bayes' Theorem \[ P(F|E) = \frac{P(F|E) \cdot P(E)}{P(F|E) \cdot P(E) + P(F'|E) \cdot P(E')}\] #### Permutations \[ P(n, k) = \frac{n!}{(n-k)!} \] #### Distinguishable Permutations \[ \frac{n!}{n_1! n_2! \ldots n_r!} \] #### Combinations \[ C(n, k) = \frac{n!}{k!(n-k)!} \] #### Conditional Probability \[ P(E|F) = \frac{n(E \cap F)}{n(F)} = \frac{P(E \cap F)}{P(F)} \] #### Number of Subsets - If a set has \( n \) elements, it
Expert Solution
Step 1

Introduction:

Denote E as the event that the outcome is even, and F as the event that the outcome is less than 5 or "<5", when a single die is rolled once.

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